which form a compatible triple. We will primarily be interested in the obstruction theory associated to the left most morphism, and in this case, by [1]} Proposition \(4.4\) , the perfect obstruction theory in \(D^b(\mathcal {M}(X,\tau ))\)  is given by \(R\pi _*(f^*T_X^{log}(-x_{out}))\) , where \(f\)  the universal map \(\pi : \mathfrak {C} \rightarrow \mathcal {M}(X,\tau )\) . Similar to the case \(\mathcal {M}(X,\beta )\) , we have an evaluation morphism \(ev: \mathfrak {M}^{ev}(\mathcal {X},\beta ) \rightarrow X\)  at contact order 0 points. Using these enhanced moduli stacks, given a log morphism \(W \rightarrow X\) , we take a fine and saturated fiber product \(\mathfrak {M}^{ev}(\mathcal {X},\tau )_W := \mathfrak {M}^{ev}(\mathcal {X},\tau ) \times ^{fs}_X W\)  to produce a point constrained moduli space. When \(W\)  is a trivial log point mapping to \(X\setminus D\) , this is the same as the ordinary fiber product. Similarly we define \(\mathcal {M}(X,\tau )_W := \mathcal {M}(X,\tau ) \times ^{fs}_X W\) . We have a morphism of log stacks \(\mathcal {M}(X,\tau )_W \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\tau )_W\)  pulled back from the natural morphism \(\mathcal {M}(X,\tau ) \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\tau )\) , and we equip the former morphism with a perfect obstruction theory pulled back from that of the latter morphism.
